Internal Wave Generation in Uniformly Stratified Fluids. 1 ... - LEGI

3. STATEMENT OF THE PROBLEM
3.1. Internal wave equation
In an unbounded incompressible fluid with uniform stratification, that is where the
undisturbed density ρ 0 varies exponentially with height z according to
ρ0 (z) = ρ00 e –βz , (3.1)
the buoyancy (or Brunt-Väisälä) frequency N = (gβ) 1/2 is constant. The small amplitude
internal waves generated by a mass source of strength m per unit volume are described by
the linearized equations of fluid dynamics (Brekhovskikh & Goncharov 1985 § 10.1, Lighthill
1978 § 4.1):
ρ0 ∂v
∂t
= – ∇P – ρgez , (3.2)
∇.v = m , (3.3)
∂ρ
∂t = ρ0βvz . (3.4)
Here v, P and ρ are respectively the velocity, pressure and density perturbations, and e z a
unit vector along the z-axis directed vertically upwards. Subscripts h and z will hereafter
denote horizontal and vertical components of vectors and operators.
Inferring from (3.2) that the motion is irrotational in the horizontal plane, we express v h
and P in terms of a horizontal velocity potential φ (Miles 1971), eliminate ρ by (3.4) and remark
that the resulting system of equations for v z and φ is satisfied if
with
Internal wave generation. 1. Green’s function 7
v = ∂2
∂t2 ∇∇ – βez + N2 ∇∇ h ψ , (3.5)
P = – ρ0 ∂2 ∂
+ N2 ψ , (3.6)
∂t2 ∂t
ρ = ρ0β ∂ ∂
– β ψ , (3.7)
∂z ∂t
∂ 2
∂
Δ – β
∂t2 ∂z + N2 Δh ψ = m . (3.8)